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Conductive Atomic Force Microscopy: Applied and Modeled Samuel Emmons, Dr. Doug Dunham Physics, Materials Science Introduction: University of Wisconsin-Eau Claire The Problem: What is an Atomic Force Microscope? The Solution (cont.) Qualitative vs. Quantitative: Current vs. Voltage on Hard Disk Current (A) An Atomic Force Microscope, or AFM, is a research instrument in the Scanning Probe Microscope, or SPM, family of instruments. An SPM is any device which probes or examines certain characteristics of a sample surface being studied. For example, Optical microscopes use visible light (photons) to examine surfaces and specimen. Electron Microscopes probe the sample surface with a beam of electrons to observe physical features and properties. Atomic Force Microscopes use a very sharp tip attached to a lever arm (cantilever) to probe a range of sample characteristics at the micro and nano-scale. The tip is kept very close to the surface and scans across the sample, essentially “feeling” the surface. 2.50E-008 2.00E-008 1.50E-008 1.00E-008 5.00E-009 Voltage(V) 0.00E+000 -15 -10 -5 0 5 10 -5.00E-009 Basic AFM Schematic As the tip moves back and forth across the sample a laser is reflected off of the top of the cantilever and onto a photo-detector. An electronic feedback loop works to keep constant a prescribed situation detected by the photo-detector. (See below**) A Variety of Operating Modes: **The most basic use of the AFM is to generate an image of sample topography, i.e. to see what the surface looks like. This is typically done in two modes, each of which maintains a prescribed condition on the photo-detector. One of the two, called Contact Mode, maintains a constant cantilever deflection, meaning it keeps the reflected laser beam at the same spot on the photo-detector. It does so by moving the tip/cantilever assembly up and down with a piezoelectric actuator connected to some feedback electronics. The distance the assembly moves is thus the height of the feature the tip is passing over. The other is AC, or Oscillating, Mode, in which the cantilever is driven at its natural thermal resonance frequency. The prescribed condition kept constant in this case is the amplitude of the oscillation detected by the photo-detector, and this is accomplished using the same piezoelectric actuator. -1.00E-008 -1.50E-008 -2.00E-008 15 A Comparison: While AFM topography can provide quantitative sample data, Conductive AFM provides only qualitative data. This is a difficult problem to overcome. In the constant sample voltage/scanning tip mode CAFM provides current data across the topography of the sample, so one may see what regions conduct more than others. Current vs. Voltage data is also largely qualitative. The question is how to make these measurements quantitative. A useful comparison may be made when considering the case when direct current can pass between the tip and the sample. In theory there is thus a capacitance and resistance in parallel between the tip and the sample surface in addition to the sample resistance. Therefore a parallel RC Circuit analysis is a helpful point of comparison for real data, such as the Current vs. Voltage data given in the slide to the left for a Hard Disk. Parallel RC Circuit: AC Note the similarity between this curve and the Hard Disk curve at left. The dissimilarities come from the oxide layer and amplifier effects in the real data. -2.50E-008 This is real current data from a Current vs. Voltage measurement. The +10V/-10V oscillation is applied to the sample as a triangle wave. Note how the current reaches a maximum and minimum current magnitude of 2.0E-008A, or 20nA. This is not a result of the sample but a limitation of the current signal amplifier used in the AFM. There are a couple of other issues that make analysis of this data difficult. One is the unknown combined sample and tip to sample resistance, and another is the unknown tip to sample capacitance. The curve flatness at 0V is also difficult to analyze, and it is likely due to a thin surface oxide layer. The Solution: Knowing the Unknowns: Model Parallel RC Circuit A Simple Model: In order to further extrapolate unknown variables it is helpful to have a simple model of the Conductive AFM tip over the sample. To build this, I have modeled the tip as a cone and the sample as a plane beneath. The situation in consideration is for a constant potential V0 placed on the sample surface with the tip at 0V. This is, of course, an approximation because in reality there would be a voltage drop across the sample. The target variable that we are seeking here is the tip to sample capacitance. A Boundary Value Problem A significant step forward in quantitative AFM is to narrow down and extrapolate out unknown variables. The primary unknown variables are the tip to sample capacitance (i.e. between the tip and the sample surface), the sample resistance, and the tip to sample resistance. To identify these variables it is helpful to construct some equivalent electrical circuits that model Conductive AFM operation. The two primary useful circuits are a parallel RC (Resistor/Capacitor) circuit and a series RC circuit. With this model of the tip on the sample we may attempt to calculate the capacitance of the tip/sample model. The best way to do this is to first find the potential function. A preferred method in this case is to work through a boundary valued problem in cylindrical coordinates with Laplace’s equation. The differential equation is given here: Model tip for CAFM 15 mm 4 mm Equation for RC Constant Here b1 is the value of the semi-minor axis corresponding to angular frequency u and b2 is the value of the semi minor axis corresponding to angular frequency w. Because we see that there is no polar angle dependence, the derivative with respect to phi is 0. It is also common to have a separable solution when dealing with Laplace’s equation, so that’s the solution guess: The differential equation becomes: AFM Topography/Lithography Magnetic Force Microscopy This is an image of the UWEC seal which I etched into a polycarbonate surface. The height difference from light(high) to dark(low) is about 30nm. The image seen above is the magnetic domain structure of a 16 square micron segment of a Hard Disk, taken in the Magnetic Force Mode of AFM. Current vs. Voltage on P-type Silicon When the metal coated CAFM tip is in contact with the surface a Metal Oxide Semiconductor (MOS) Capacitor is formed, completing a series RC Circuit. Series RC Current vs. Voltage Curve Conductive Atomic Force Microscopy : Topography of Hard Disk A Hard Disk is a good test sample for CAFM because it is highly conductive. This image is a little less than 3 microns wide, and has about a 5nm range in height. The color was added to the image to increase contrast. Conductive Atomic Force Microscopy, or CAFM, refers to the mode of AFM operation in which a current is passed through the tip/sample connection to determine the conductive properties of a sample surface. There are two relatively easy ways of determining these conductive properties. One is to maintain a constant voltage on the sample and move the tip, resulting in a variety of currents measured dependent on tip location on the sample. The other is by keeping the tip position constant and varying the voltage on the surface, thus obtaining current vs. voltage data for a specific point on the sample. The general solutions for R(r) and Z(z) are: Here J0 is a Bessel Function of the first kind, and Y0 is a Bessel Function of the second kind. I have yet to apply the boundary conditions (given potential info.) to this solution. When the tip and sample connection cannot pass direct current between them, the circuit acts as a series RC circuit, where the resistance is the sample resistance and the capacitance is the tip to sample capacitance. The above curve was made in Maple. First, I solved the associated differential equation for charge as a function of time, Q(t). The differential equation is: b2 Future Development: Complete Model: Completing the model will allow for the implementation of the theory to a real sample. This sample would likely be a cleaned Silicon substrate with approximately known sample resistance, so that the accuracy of the capacitance model may be determined. R*(dQ(t)/dt)+Q(t)/C = V*cos(w*t) where w is the angular frequency at which the sample voltage is driven. Taking the time derivative of the solution Q(t) gives the current, I(t), which, plotted against voltage, yields the above curve. Noticing that this is an ellipse, it turns out that knowing the value of the semi-major and semi-minor axes for two values of w allows one to solve for the value of R multiplied by C, or RC. This model was created in Maple. As is evident, the tip has been modeled as a cone and the surface as an infinite plane. b1 Sample Application: Once the model is finessed, it may be applied to certain fabricated samples to determine if the electrical properties of the actual sample closely match expected values. RC Value on P-type Silicon w=4pHz; u=2pHz; b1=8.1E-11A; b2=1.49E-10A RC=0.159s approximately. Acknowledgements: UWEC Materials Science Center